The Thick-Thin Decomposition
Talk by Ken Bromberg
August 20, 2007
This lecture discusses the Thick-Thin decomposition for hyperbolic manifolds. The starting
point for this discussion is the Margulis lemma. While the hypotheses of the Margulis
Lemma are rather general, for hyperbolic space the result can be stated as follows:
Lemma: There exists a number n > 0 such that if Γ is a discrete group of isometries of
Hn generated by γ1 , γ2 , ..., γk and there exists p ∈ H3 such that d(p, γi p) ≤ n , then Γ is
virtually nilpotent.
A group is virtually “X” if it has a finite index subgroup which is “X”. A nilpotent group
of hyperbolic isometries is abelian, so we can replace “nilpotent” in the conclusion of the
theorem with “abelian,” and in dimensions 2 and 3, the cases that concern us here, it turns
out that we don’t need to pass to a finite index subgroup. So the Margulis lemma tells us
that groups of hyperbolic isometries generated by elements with small enough translation
length are abelian. For M a hyperbolic manifold, let I(x) denote the injectivity radius at
x, and let M < denote the set {x ∈ M | I(x) < }. As a corollary to the Margulis Lemma
we have
Corollary: If X is a hyperbolic surface, then each component of X <2 is either an
annulus or a cusp.
Annulus
Cusp
Path of length
c < 2ε2
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Proof. Let X = H2 /Γ, Γ ⊂ Isom(H2 ) an infinite discrete group without elliptic elements.
Define Γp, < Γ to be h{γ ∈ Γ | d(p, γ(p)) < 2 }i. By Margulis’ Lemma, Γp,2 is abelian,
and because commuting elements in a Fuchsian group share fixed points, Γp,2 must be
generated by a single element. The two cases depend on whether the generating element
is hyperbolic or parabolic. The following picture shows the hyperbolic case.
c < ε2
γp
γ
p
By continuity,
there exist segments
of length ε2
Paths of
length
ε2
Cover corresponding to <γ>
Universal Cover
It is an exercise to check that if the tube between the paths of length 2 did not embed in
the quotient surface, then there would be another loop of length less than 2 , contradicting
the Margulis lemma.
The parabolic case is almost identical, and is left as an exercise.
The following corollary follows immediately from the above discussion.
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Corollary: If X has genus g, then there are at most 3g − 3 closed geodesics of length
less than 22 .
The picture is roughly the same in dimension three, as the following theorem shows.
Theorem: The components of M <3 are of three types:
1. r-neighborhoods of simple closed geodesics of length less than 3
2. Quotients of horoballs by a parabolic subgroup isomorphic to Z
3. Quotients of horoballs by a subgroup generated by two parabolics isomorphic to Z⊕Z
To prove this one uses the same argument as above. By Margulis’ Lemma, Γp,3 is an
abelian group, so by considering fixed points as above, one can show that it is generated
either by a single loxodromic element, a single parabolic element, or two parabolic elements
that share a fixed point at infinity. These give the three cases above.
Both the topology and geometry of the thin parts in dimension two were very easy to
understand. While the above theorem tells us the topology of the thin parts in dimension
three, there are some subtleties in figuring out what these tubes can look like geometrically.
Consider first case 2 above, which is called a rank-1 cusp. Topologically, a rank-1 cusp is
homeomorphic to to the product of an annulus A and the real line. Conjugating the fixed
point to be at infinity in the upper half space model, we get the following picture.
Level Annulus
As the above picture shows, annuli correspond to strips of horospheres glued together, and
the intrinsic metric on these horizontal planes is a Euclidean metric dE 2 . As we go up
towards infinity, the Euclidean metric on the annuli foliating the cusp is contracted, so
that the metric looks like e−2t dE 2 + dt2 . We should think of the cusp as being modeled
on A × (0, ∞) rather than A × R, as the closure of one end of the cusp does not lie at
infinity, but sits in hyperbolic space. There is a unique parabolic element in Isom+ (H3 ) up
to conjugacy, so all rank-1 cusps are isomorphic.
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Case 3 above, the case of a rank-2 cusp, is similar. Conjugating the shared fixed points
of the parabolics to infinity, we get that the restriction of the action of this group to the
horizontal planes foliating H3 is a discrete Z ⊗ Z action by translations. The quotient of
each plane is a torus, and the intrinsic metric on this torus is again Euclidean, contracting
like e−2t dE 2 + dt2 as we move out the cusp.
Level Torus
The geometry of the cusp is determined by the Euclidean structure of the foliating tori.
The final case is the most intricate, and is the only case where the thin part is compact.
A simple closed geodesic corresponds to the axis of a loxodromic element in the univeral
cover. Conjugating the loxodromic element to fix zero and infinity, the r-neighborhood of
this element is a cone. The hyperbolic trigonometry of this cone is worked out below.
r
coth(r)
sinh(r)
l
tanh(r)
cosh(l)
r
The labels on the left hand side of the diagram show the directions of principal curvature,
which lie in the vertical plane through the origin and in the horizontal plane. Multiplying
the principal curvatures together and adding the curvature of the ambient space, we see
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that this cone is flat. A loxodromic element translates along its axis, so a fundamental
domain is a conical wedge and after identification the boundary is a flat torus. A loxodromic
element also twists around its axis, so closed geodesics on the boundary may wind around
the conical wedge.
Closed Geodesic
on boundary
Note that on the boundary of the solid torus there is a unique curve (up to isotopy) that
bounds a disk when the torus is filled in. Such a curve is called the meridian of the torus.
Given any Euclidean torus T and any essential closed curve γ on T , there is a loxodromic
tube whose boundary is isometric to T and on which γ is the meridian.
To construct such a tube, take the desired torus T and the desired curve γ and cut along
γ to get a cylinder. There exists a cone C whose circumference matches the length of γ
(as indicated above, such a cone will be the set of points at distance arcsinh(l(γ)/2π)). We
then find a loxodromic element λ that identifies circles on the cone whose distance apart is
equal to the height of the cylinder, and such that the shortest curve on C/λ has the same
length as the shortest on T .
1: Cut along desired
meridian γ
Shortest
closed curve
3: Find a geodesic on the
boundary whose length is
equal to the shortest closed
curve on the original torus
γ
f(γ)
γ
h
h
2: Embed on a cone
so that f(γ) has the
right length
f
γ’
γ
h
f(γ’)
4: Find a loxodromic element
that has translation length h
and twists so that the endpoints of the shortest curve
match up
Now consider the effect of taking the n1 curve γn and making it a meridian. If n is large,
then h in the above picture is very short. We therefore get a fundamental domain for the
action that is short and deep, as the distance from the boundary torus to the axis must
become large. Given a solid torus S with meridian γn and a rank-2 cusp C such that ∂S is
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isometric to ∂S, we can map a chunk of the C into S by mapping boundary to boundary,
and sending foliating tori further out the cusp to foliating tori deeper in the solid torus.
While this map is not defined on the whole cusp and is only an isometry on the boundary,
ine can check that as n gets bigger and the tube gets deeper, this map can be defined
further out the cusp, and approaches an isometry where it is defined.
Thus the rank two cusp arises as a limit of deeper and deeper tubes. Let λn be the
loxodromic element used to define the solid torus with meridian γn . As n gets large, γn
lies more and more in the horizontal plane, and therefore λn converges to a parabolic.
One might suppose, then, that as the solid torus is glued up using λn , the “limit” of
this sequence of spaces should be a rank one cusp corresponding to λ∞ . This is not the
case, however, as λnn corresponds to translation up the cone, and in the limit gives a second
parabolic element. This is an instance of “algebraic limits” and “geometric limits” differing,
as looking at the limiting behavior of generators of a Kleinian group and looking at the
limiting geometric picture give different results. This will be described in more detail in a
later lecture.
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